Quasinorm


In linear algebra, functional analysis and related areas of mathematics, a quasinorm is similar to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced by
for some.

Related concepts


  1. Non-negativity: ;
  2. Absolute homogeneity: for all and all scalars ;
  3. there exists a such that for all.
If is a quasinorm on then induces a vector topology on whose neighborhood basis at the origin is given by the sets:
as ranges over the positive integers.
A topological vector space with such a topology is called a quasinormed space.
Every quasinormed TVS is a pseudometrizable.
A vector space with an associated quasinorm is called a quasinormed vector space.
A complete quasinormed space is called a quasi-Banach space.
A quasinormed space is called a quasinormed algebra if the vector space A is an algebra and there is a constant K > 0 such that
for all.
A complete quasinormed algebra is called a quasi-Banach algebra.

Characterizations

A topological vector space is a quasinormed space if and only if it has a bounded neighborhood of the origin.